p-variation of strong Markov processes

Abstract

Let ξ_t, t\in[0,T], be a strong Markov process with values in a complete separable metric space (X,ρ) and with transition probability function P_{s,t}(x,dy), 0\le s\le t\le T, x\in X. For any h\in[0,T] and a>0, consider the function α(h,a)=sup\bigl{P_{s,t}\bigl(x,{y:ρ(x,y)\ge a}\bigr):x\in X,0\le s\le t\le (s+h)\wedge T\bigr}. It is shown that a certain growth condition on α(h,a), as a\downarrow0 and h stays fixed, implies the almost sure boundedness of the p-variation of ξ_t, where p depends on the rate of growth.